Portal de Periódicos da CAPES

Existence and multiplicity of nontrivial solutions to the modified Kirchhoff equation without the growth and Ambrosetti–Rabinowitz conditions

2021; University of Szeged; Issue: 83 Linguagem: Inglês

10.14232/ejqtde.2021.1.83

1417-3875

Zhongxiang Wang, Gao Jia,

Stability and Controllability of Differential Equations

The paper focuses on the modified Kirchhoff equation \ b e g i n { a l i g n * } - \ l e f t ( a + b \ i n t _ { \ m a t h b b { R } ^ N } | \ n a b l a u | ^ 2 d x \ r i g h t ) \ D e l t a u - u \ D e l t a ( u ^ 2 ) + V ( x ) u = \ l a m b d a f ( u ) , \ q u a d x \ i n \ m a t h b b { R } ^ N , \ e n d { a l i g n * } w h e r e $ a , b & g t ; 0 $ , $ V ( x ) \ i n C ( \ m a t h b b { R } ^ N , \ m a t h b b { R } ) $ a n d $ \ l a m b d a & l t ; 1 $ i s a p o s i t i v e p a r a m e t e r . W e j u s t a s s u m e t h a t t h e n o n l i n e a r i t y $ f ( t ) $ i s c o n t i n u o u s a n d s u p e r l i n e a r i n a n e i g h b o r h o o d o f $ t = 0 $ and at infinity. By applying the perturbation method and using the cutoff function, we get existence and multiplicity of nontrivial solutions to the revised equation. Then we use the Moser iteration to obtain existence and multiplicity of nontrivial solutions to the above original Kirchhoff equation. Moreover, the nonlinearity f ( t ) may be supercritical.